Integrand size = 27, antiderivative size = 123 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \]
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Time = 0.35 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2953, 3055, 3060, 2852, 212} \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a \sin (c+d x)+a}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a \sin (c+d x)+a}}+\frac {2 a \cos (c+d x) \sqrt {a \sin (c+d x)+a}}{5 d}+\frac {2 \cos (c+d x) (a \sin (c+d x)+a)^{3/2}}{5 d} \]
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Rule 212
Rule 2852
Rule 2953
Rule 3055
Rule 3060
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc (c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{5/2} \, dx}{a^2} \\ & = \frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {2 \int \csc (c+d x) (a+a \sin (c+d x))^{3/2} \left (\frac {5 a^2}{2}-\frac {3}{2} a^2 \sin (c+d x)\right ) \, dx}{5 a^2} \\ & = \frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+\frac {4 \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \left (\frac {15 a^3}{4}+\frac {3}{4} a^3 \sin (c+d x)\right ) \, dx}{15 a^2} \\ & = -\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}+a \int \csc (c+d x) \sqrt {a+a \sin (c+d x)} \, dx \\ & = -\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d}-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d} \\ & = -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \cos (c+d x)}{\sqrt {a+a \sin (c+d x)}}\right )}{d}-\frac {2 a^2 \cos (c+d x)}{5 d \sqrt {a+a \sin (c+d x)}}+\frac {2 a \cos (c+d x) \sqrt {a+a \sin (c+d x)}}{5 d}+\frac {2 \cos (c+d x) (a+a \sin (c+d x))^{3/2}}{5 d} \\ \end{align*}
Time = 5.42 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.18 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {(a (1+\sin (c+d x)))^{3/2} \left (5 \cos \left (\frac {3}{2} (c+d x)\right )-\cos \left (\frac {5}{2} (c+d x)\right )-10 \log \left (1+\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+10 \log \left (1-\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+5 \sin \left (\frac {3}{2} (c+d x)\right )+\sin \left (\frac {5}{2} (c+d x)\right )\right )}{10 d \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3} \]
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Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {2 \left (1+\sin \left (d x +c \right )\right ) \sqrt {-a \left (\sin \left (d x +c \right )-1\right )}\, \left (5 a^{\frac {5}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (d x +c \right )}}{\sqrt {a}}\right )-\left (a -a \sin \left (d x +c \right )\right )^{\frac {5}{2}}+5 a \left (a -a \sin \left (d x +c \right )\right )^{\frac {3}{2}}-5 a^{2} \sqrt {a -a \sin \left (d x +c \right )}\right )}{5 a \cos \left (d x +c \right ) \sqrt {a +a \sin \left (d x +c \right )}\, d}\) | \(123\) |
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Leaf count of result is larger than twice the leaf count of optimal. 282 vs. \(2 (105) = 210\).
Time = 0.27 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.29 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\frac {5 \, {\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} - 9 \, a \cos \left (d x + c\right ) + {\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) - 4 \, {\left (a \cos \left (d x + c\right )^{3} - 2 \, a \cos \left (d x + c\right )^{2} - 2 \, a \cos \left (d x + c\right ) - {\left (a \cos \left (d x + c\right )^{2} + 3 \, a \cos \left (d x + c\right ) + a\right )} \sin \left (d x + c\right ) + a\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{10 \, {\left (d \cos \left (d x + c\right ) + d \sin \left (d x + c\right ) + d\right )}} \]
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Timed out. \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \cos \left (d x + c\right )^{2} \csc \left (d x + c\right ) \,d x } \]
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Time = 0.43 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.34 \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=-\frac {\sqrt {2} {\left (16 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 40 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, \sqrt {2} a \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) + 20 \, a \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )} \sqrt {a}}{10 \, d} \]
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Timed out. \[ \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^{3/2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{3/2}}{\sin \left (c+d\,x\right )} \,d x \]
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